Find the solution of the following equation whose argument is strictly between $180^\circ$ and $270^\circ$. Round your answer to the nearest thousandth. $z^5=-243i$ $z$ =
The Strategy A straightforward way to solve an equation of the form $z^{n}=m$ is by using the polar form of $z$. Therefore, our solution will consist of the following steps: Rewrite $z^n$ and $m$ in polar form. [How is this done, in general?] Solve for the modulus and argument of $z$. Find the rectangular form of $z$. [How is this done, in general?] Rewrite the equation in polar form Let's denote $r$ and $\theta$ to be the modulus and argument of $z$, respectively. Therefore, $z^{5}=r^{5}[\cos {({5}\cdot\theta)}+i\sin {({5}\cdot\theta)}]$. The number $-243i$ has a modulus of $243$. The argument of $-243i$ can be $270^\circ$ plus any multiple of $360^\circ$, so we can write it as $270^\circ+k\cdot360^\circ$ for an integer $k$. Now the equation looks as follows: $\begin{aligned}r^{5}[\cos {({5}\cdot\theta)}+i\sin {({5}\cdot\theta)}]&= \\243&[\cos(270^\circ+k\cdot360^\circ)+i\sin(270^\circ+k\cdot360^\circ)]]\end{aligned}$ When two complex numbers are equal, we know that both their moduli and arguments are equal. Therefore, we have the following equations for $r$ and $\theta$ : $r^{5}=243$ ${5}\cdot\theta=270^\circ+k\cdot360^\circ$ Solving for $r$ $\begin{aligned}r^{5}&=243 \\\\ r &=3 \end{aligned}$ Solving for $\theta$ $\begin{aligned}{5}\cdot\theta&=270^\circ+k\cdot360^\circ \\\\\theta&=54^\circ+k\cdot72^\circ\end{aligned}$ Remember that $\theta$ is strictly between $180^\circ$ and $270^\circ$. Therefore, we need to find the multiple of $72^\circ$ that is strictly within the range of $180^\circ-54^\circ=126^\circ$ and $270^\circ-54^\circ=216^\circ$. This multiple is simply $144^\circ$, so $\theta=198^\circ$. Finding the rectangular form of $z$ Let's plug in $r=3$ and $\theta=198^\circ$ into the polar form of $z$ : $\begin{aligned}z&=r[\cos(\theta)+i\cdot\sin(\theta)]\\\\ &=3[\cos(198^\circ)+i\cdot\sin(198^\circ)]\\\\ &=3\cos(198^\circ)+3\sin(198^\circ)\cdot i\end{aligned}$ Using the calculator and rounding to the nearest thousandth, we get the following solution: $z=-2.853-0.927i$ Summary $z=-2.853-0.927i$